Numerical methods for non-standard fractional operators in the simulation of dielectric materials

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1 Numerical methods for non-standard fractional operators in the simulation of dielectric materials Roberto Garrappa Department of Mathematics - University of Bari - Italy roberto.garrappa@uniba.it Fractional PDEs: Theory, Algorithms and Applications ICERM - Providence, June 18 22, 2018 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

2 Outline Numerical methods for non-standard fractional operators in the simulation of dielectric materials 1 The problem 2 The operator 3 The numerical method Financial support: EU Cost Action Fractional Systems Istituto Nazionale di Alta Matematica (INdAM-GNCS) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

3 Maxwell s equations: Standard Maxwell s equations: H = ɛ 0 t E H E = µ 0 t E: electric field H: magnetic field Ampere s law Faraday s law R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

4 Maxwell s equations: Standard Maxwell s equations: H = ɛ 0 t E H E = µ 0 t E: electric field H: magnetic field Ampere s law Faraday s law Real world applications design of data and energy storage devices design of antennas medical diagnostic (MRI), cancer therapy,... R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

5 Maxwell s equations: Standard Maxwell s equations with polarization: H = ɛ 0 t E + t P H E = µ 0 t Ampere s law Faraday s law E: electric field H: magnetic field P: polarization R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

6 Maxwell s equations: Standard Maxwell s equations with polarization: H = ɛ 0 t E + t P H E = µ 0 t Ampere s law Faraday s law E: electric field H: magnetic field P: polarization The complex susceptibility The polarization P depends on the electric field E (constitutive law) ˆP = ε 0 ˆχ(ω) Ê ˆχ(ω) is a specific feature of the matter (or system) Simplified notation (just for easy of presentation) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

7 Determining the complex susceptibility ˆχ(ω) How to derive ˆχ(ω) = ˆχ (ω) iˆχ (ω)? R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

8 Determining the complex susceptibility ˆχ(ω) How to derive ˆχ(ω) = ˆχ (ω) iˆχ (ω)? Experimental data (in the frequency domain): ˆP = ˆχ(ω)Ê 10 0 Exper. data 10 1 ˆχ (ω) frequency ω Match experimental data into a mathematical model R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

9 Determining the complex susceptibility ˆχ(ω) 10 0 Exper. data Debye model 10 1 ˆχ (ω) frequency ω The Debye model: ˆχ(ω) = iω (standard materials) Ordinary differential equation: d P(t) + P(t) = E(t) dt R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

10 Determining the complex susceptibility ˆχ(ω) 10 0 Exper. data 10 1 ˆχ (ω) frequency ω Materials with anomalous dielectric properties: amorphous polymers complex systems (biological tissues) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

11 Determining the complex susceptibility ˆχ(ω) 10 0 Exper. data Debye model 10 1 ˆχ (ω) frequency ω Debye model not satisfactory R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

12 Determining the complex susceptibility ˆχ(ω) Exper. data Debye model C C model ˆχ (ω) frequency ω The Cole-Cole model: ˆχ(ω) = Fractional differential equation: Only partially satisfactory (iω) α 0 < α < 1 α d α P(t) + P(t) = E(t) dtα R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

13 Determining the complex susceptibility ˆχ(ω) Exper. data Debye model C C model H N model ˆχ (ω) frequency ω The Havriliak-Negami model: ˆχ(ω) = Better matching thanks to three parameters α, γ and 1 ( 1 + (iω) α ) γ 0 < α, αγ 1 S. Havriliak and S. Negami A complex plane representation of dielectric and mechanical relaxation processes in some polymers. In: Polymer (1967) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

14 Determining the complex susceptibility ˆχ(ω) Exper. data Debye model C C model H N model ˆχ (ω) frequency ω The Havriliak-Negami model: ˆχ(ω) = 1 ( 1 + (iω) α ) γ 0 < α, αγ < 1 Fractional pseudo-differential equation: (1 + α d α ) γ P(t) = E(t)? dt α R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

15 Other models for complex susceptibility ˆχ(ω) Modified Havriliak-Negami or JWS (Jurlewicz, Weron and Stanislavsky) ˆχ JWS (iω) = 1 1 (1 + ( i ω ) α ) γ = 1 (i ω) αγ χ HN (iω) EW: Excess wing (Hilfer, Nigmatullin and others) ˆχ EW (iω) = Multichannel excess wing (Hilfer) 1 + ( 2 iω) α 1 + ( 2 iω) α + 1 iω. 1 ˆξ(s) = [ n 1 + (iω k ) α k k=1 This talk focuses on the Havriliak-Negami model ] 1 Garrappa R., Mainardi F. and Maione G., Models of dielectric relaxation based on completely monotone functions. In: Frac. Calc. Appl. Anal. 19(5) (2016) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

16 Dealing with the Havriliak-Negami model ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) Few contributions on simulation of this constitutive law: C.S.Antonopoulos, N.V.Kantartzis, I.T.Rekanos FDTD Method for Wave Propagation in Havriliak-Negami Media Based on Fractional Derivative Approximation. In: IEEE Trans Magn. 53(6) (2017) P.Bia et al. A novel FDTD formulation based on fractional derivatives for dispersive Havriliak Negami media. In: Signal Processing, 107 (2015) M.F.Causley, P.G.Petropoulos, On the Time-Domain Response of Havriliak-Negami Dielectrics. In: IEEE Trans. Antennas Propag., 61(6) (2013) M.F.Causley, P.G.Petropoulos, and S. Jiang Incorporating the Havriliak-Negami dielectric model in the FD-TD method. In: J. Comput. Phys., 230 (2011), Main problems: Define time-domain operator for HN Discretize the operator for simulations R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

17 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

18 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Combination of fractional operators ( α 0 Dt α + 1 ) γ = exp ( t ) α α 0 Dt 1 α αγ 0D αγ t ( t ) exp α α 0 Dt 1 α Useful for theoretical investigations R.R.Nigmatullin and Y.E.Ryabov Cole Davidson dielectric relaxation as a self similar relaxation process. In: Physics of the Solid State 39.1 (1997) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

19 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Expansion in infinite series ( α 0 D α t + 1 ) γ = k=0 ( ) γ α(γ k) 0D α(γ k) t k No satisfactory for error control V.Novikov et al. Anomalous relaxation in dielectrics. Equations with fractional derivatives. In: Mater. Sci. Poland 23.4 (2005) P.Bia et al. A novel FDTD formulation based on fractional derivatives for dispersive Havriliak Negami media. In: Signal Processing, 107 (2015) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

20 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Prabhakar derivative ( α 0 D α t + 1 ) γ P(t) = d dt t 0 ( u Prabhakar function: E γ α,β (z) = 1 Γ(γ) ) αγ ( ( E γ u ) α ) α,1 αγ P(t u) du k=0 Γ(γ + k)z k k!γ(αk + β) R.Garra, R.Gorenflo, F.Polito and Z.Tomovski Hilfer-Prabhakar derivatives and some applications. In: Appl. Math. Comput. 242 (2014) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

21 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Prabhakar derivative of Caputo type ( α 0 Dt α + 1 ) γ d t ( u ) αγ ( ( P(t) = E γ u ) α ) α,1 αγ P(t u) du dt 0 RL C( α 0 Dt α + 1 ) t ( γ u P(t) = 0 ) αγ ( ( E γ u ) α ) α,1 αγ P (t u) du Caputo C( α 0 D α t + 1 ) γ P(t) = ( α 0 D α t + 1 ) γ ( P(t) P(0 + ) ) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

22 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Fractional differences of Grünwald Letnikov ( α 0 Dt α + 1 ) γ 1 P(t) = lim h 0 h αγ Ω k P(t kh) k=0 Ω 0 = 1, Ω k = α α + h α k ( (1 + γ)j j=1 k ) 1 ω (α) j Ω k j, Ω k = ( α + h α)γ Ω k R.Garrappa On Grünwald Letnikov operators for fractional relaxation in Havriliak-Negami models. In: Commun. Nonlinear. Sci. Numer. Simul. 38 (2016) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

23 Operators in the time domain ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) ( α 0 D α t + 1 ) γ P(t) = E(t)??? How to define the fractional pseudo-differential operator ( α 0D α t + 1 ) γ? Fractional differences of Grünwald Letnikov ( α 0 Dt α + 1 ) γ P(tn ) 1 n h αγ Ω k P(t n k ) + O ( h ) C( α 0 D α t + 1 ) γ P(tn ) 1 h αγ k=0 n ( Ω k P(tn k ) P(t 0 ) ) + O ( h ) k=0 Approximation of first order only! R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

24 Integral operator ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) P(t) = ( α 0J α t + 1 ) γ E(t) Prabhakar integral ( α 0J α t P(t) = 1 αγ t ) γ (t u) αγ 1 E γ α,αγ ( ( t u ) α ) E(u) du, R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

25 Integral operator ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) P(t) = ( α 0J α t + 1 ) γ E(t) Prabhakar integral ( α 0J α t P(t) = 1 αγ t ) γ (t u) αγ 1 E γ α,αγ ( ( t u ) α ) E(u) du, Non-local operator Weakly singular integral Series of RL integrals 1 ( α 0J α t + 1 ) γ ( ) γ 1 = ( 1) k k k=0 Jα(k+γ) α(k+γ) 0 1 Giusti A., A comment on some new definitions of fractional derivative. Nonlinear Dyn. (2018) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

26 Integral operator ˆP(ω) = 1 ((iω) α + 1) γ Ê(ω) P(t) = ( α 0J α t + 1 ) γ E(t) Prabhakar integral ( α 0J α t P(t) = 1 αγ t ) γ (t u) αγ 1 E γ α,αγ ( ( t u ) α ) E(u) du, Preferable in numerical simulation Numerical integration is easier than numerical differentiation Approximation by quadrature rules What kind of quadrature rule? R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

27 Discretization of Maxwell s equations H = ɛ 0 t E + t P H E = µ 0 t Ampere s law Faraday s law F = ( Fz y F ) ( y Fx i + z z F ) ( z Fy j + x x F ) x k y The Finite Differences Time Domain (FDTD) method: 1 Introduced by K. Yee in Approximation based on centred finite difference operators 3 Use of staggered grids in space and time R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

28 Discretization of Maxwell s equations - FDTD t n+2 t n+1 t n Ampere s law Faraday s law t n 1 t n 2 x m 2 x m 1 x m x m+1 x m+1 H n+ 1 2,m = H m+ 1 2,n+ 1 2 H m 1 2,n+ 1 2 x + O ( 2 ) x E m+ 1 2,n = E m+1,n H m 1,n + O ( 2 ) x x R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

29 Discretization of Maxwell s equations - FDTD t n+2 t n+1 t n E H t n 1 t n 2 x m 2 x m 1 x m x m+1 x m+1 H n+ 1 2,m = H m+ 1 2,n+ 1 2 H m 1 2,n+ 1 2 x + O ( 2 ) x E m+ 1 2,n = E m+1,n H m 1,n + O ( 2 ) x x R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

30 Discretization of the constitutive law H = ɛ 0 t E + t P E = µ 0 H t Ampere s law Faraday s law P = ( α 0J α t + 1 ) γ E Constitutive law It is necessary a second-order approximation Use of the values of E just on grid points (t n, x m ) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

31 Discretization of the constitutive law H = ɛ 0 t E + t P E = µ 0 H t Ampere s law Faraday s law P = ( α 0J α t + 1 ) γ E Constitutive law It is necessary a second-order approximation Use of the values of E just on grid points (t n, x m ) Convolution quadratures by Lubich [Lubich, 1988] Based on the LT ˆχ(s) = 1 ( α s α + 1 ) γ (i.e. the complex susceptibility) Extension of classical linear multistep methods (LMM) for ODEs Preservation of convergence and stability of the LMM R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

32 Convolution quadrature rules I (t) = t Generalizes LMM for ODEs k ρ j y n j = h j=0 0 f (t u)g(u)du k σ j f (t n j, y n j ), n k. j=0 Based on the generating function of the LMM ρ(ξ) = k ρ j ξ k j σ(ξ) = j=0 I (t n ) = h µ k σ j ξ k j j=0 ν n w n,j g(t j ) + h µ ω n j g(t j ) j=0 } {{ } starting term j=0 } {{ } convolution term δ(ξ) = ρ(1/ξ) σ(1/ξ) Lubich C. Convolution quadrature and discretized operational calculus (I and II), In: Numer. Math. 52, and (1988). R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

33 Convolution quadrature rules I (t) = t 0 ν n f (t u)g(u)du = I (t n ) = h µ w n,j g(t j ) + h µ ω n j g(t j ) j=0 } {{ } starting term j=0 } {{ } convolution term Use of LT F (s) of f (t) F (s) analytic in a sector Σ ϕ,c = { s C : arg(s c) < π ϕ } F (s) M s µ µ > 0 M < ( ) δ(ξ) Weights F = ω n ξ n h n=0 ϕ c Lubich C. Convolution quadrature and discretized operational calculus (I and II), In: Numer. Math. 52, and (1988). R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

34 Convolution quadrature for Havriliak-Negami I (t) = t 0 ν n f (t u)g(u)du = I (t n ) = h µ w n,j g(t j ) + h µ ω n j g(t j ) j=0 } {{ } starting term j=0 } {{ } convolution term ( f (t) = t αγ 1 Eα,αγ γ t α / α) 1 F (s) = ˆχ(s) = ( α s α + 1) γ F (s) analytic outside (, 0] F (s) M s αγ µ = αγ ( ) δ(ξ) Weights ˆχ = ω n ξ n h n=0 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

35 Convolution quadrature for Havriliak-Negami P = ( α 0J α t + 1 ) γ E P(x, t n ) = h αγ ν n w n,j E(x, t j ) + h αγ ωn HN E(x, t j ) j=0 } {{ } starting term Trapezoidal rule y n+1 y n = h 2 Generating function δ(ξ) = Convolution weights h αγ n=0 2(1 ξ) 1 + ξ j=0 } {{ } convolution term ( f (tn, y n ) + f (t n+1, y n+1 ) ) ( ) 2(1 ξ) ωn HN ξ n = ˆχ h(1 + ξ) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

36 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 1) Recursion, FFT and power of FPS 2) Numerical integration of Cauchy integral R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

37 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 1) Recursion, FFT and power of FPS ( h α + (1 ξ) α = (1 + ξ) α = ˆω n ξ n ˆω 0 = 1 ˆω n = n=0 ( 1 α + 1 n ( ) 1 α ω n ξ n ω 0 = 1 ω n = 1 n n=0 ) ˆω n 1 ω n 1 (1 ξ) α (1 + ξ) α = ω n ξ n FFT of weights of (1 ξ) α and (1 + ξ) α n=0 ) γ ( ) γ ω n ξ n 1 ω n = (1 + h 1 + α ) γ 1 + h ω nξ n unitary FPS α n=0 n=1 R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

38 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 1) Recursion, FFT and power of FPS Miller s Formula: power β C of unitary Formal Power Series ( 1 + a1 ξ + a 2 ξ 2 + a 3 ξ ) β (β) = v 0 + v (β) 1 ξ + v (β) 2 ξ 2 + v (β) 3 ξ n ( ) v (β) (β + 1)j 0 = 1, v n (β) = 1 a j v (β) n j n. j=1 Direct evaluation ω HN 0 = 1 ω HN n = n ( (1 γ)j j=1 n ) 1 ω j ωn j HN (1 ξ) α (1+ξ) α R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

39 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 1) Recursion, FFT and power of FPS Weights evaluated in an exact way Expensive computation O(N 2 ) Instability due to long recurrences R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

40 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 2) Numerical integration of Cauchy integral ω HN n = 1 n! d n dξ n G(ξ) G(ξ) = ( h α + ξ=0 Representation in terms of Cauchy integrals ωn HN = 1 ξ n 1 G(ξ)dξ, 2πi Approximation by quadrature rule C ) γ (1 ξ)α (1 + ξ) α R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

41 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 2) Numerical integration of Cauchy integral Contour ξ = ρ 0 < ρ < 1 0 < ρ < 1 Equispaced quadrature nodes ξ l = ρe 2πil/L, l = 0, 1,..., L Trapezoidal rule ωn,l HN = ρ n L 1 G(ξ l )e 2πinl/L L l=0 Fast computation by FFT O(L log L) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

42 Evaluation of convolution weights h αγ n=0 ) γ ωn HN ξ n = ( h hαγ α (1 ξ)α h 2 α α + (1 + ξ) α, h = 2 2) Numerical integration of Cauchy integral Choice of ρ and L: error analysis 0 < ρ < Discretization error 0 < ρ < r < 1 ω n HN ωn,l HN ρ n (r/ρ) n a C r (r/ρ) L 1 Round-off error ω HN n,l ˆω n,l HN ρ n C ρ ɛ Balancing of the two errors: target accuracy ε > ɛ a Trefethen & Weideman in [SIAM Review 2014] R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

43 Numerical integration of Cauchy integral 10 8 Error α=0.8 γ=0.9 = 10 2 h= n 10 9 Error α=0.7 γ=0.8 = 10 3 h= n R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

44 Evaluation of the starting term P(x, t n ) = h αγ ν n w n,j E(x, t j ) + h αγ ωn HN E(x, t j ) j=0 } {{ } starting term j=0 } {{ } convolution term Handling the singularity at the origin w n,j obtained by imposing the rule exact for {1, t αγ, t} when αγ > 1 2 Solution of small systems of algebraic equations n n αγ E γ α,αγ αγ w ( (t n/) α ) j=0 n,0 w n,1 = γ α n 2αγ E γ α,2αγ+1 ( (t n j n/) α ) w n,2 j=0 n n αγ+1 E γ α,αγ+2 ( (t n/) α ) γ α = Γ(αγ + 1) Simple inversion of the matrix j=0 ω HN n j ω HN n j αγ ω HN n jj R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

45 Evaluation of Prabhakar (three-parameter ML) function Matlab code available Possible evaluation of E γ α,β (z) when arg(z) > απ Also available code for ML function with matrix arguments R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

46 Testing the quadrature rule N N P(t) = ( α 0Jt α + 1) γ E(t) E(t) = a k cos ω k t + b k sin ω k t k=0 k=0 4 α γ Set Set Set α=0.9 γ=0.9 = α=0.8 γ=0.8 =10 0 α=0.8 γ=0.7 = t R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

47 Numerical experiments: errors and EOC N N P(t) = ( α 0Jt α + 1) γ E(t) E(t) = a k cos ω k t + b k sin ω k t k=0 k=0 α = 0.9 γ = 0.9 α = 0.8 γ = 0.8 α = 0.8 γ = 0.7 = 10 2 = 10 0 = 10 1 h Error EOC Error EOC Error EOC ( 3) 6.21( 3) 2.04( 3) ( 3) ( 3) ( 4) ( 4) ( 4) ( 4) ( 5) ( 5) ( 5) ( 5) ( 5) ( 6) ( 6) ( 6) ( 6) EOC = log 2 ( E(h)/E( h 2 )) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

48 Numerical experiments: errors and EOC N N N P(t) = ( α 0Jt α + 1) γ E(t) E(t) = a k t αk + b k t γk + c k t k k=0 k=0 k=0 α = 0.9 γ = 0.9 α = 0.8 γ = 0.8 α = 0.8 γ = 0.7 = 10 2 = 10 0 = 10 1 h Error EOC Error EOC Error EOC ( 4) 1.26( 3) 1.43( 4) ( 5) ( 4) ( 5) ( 6) ( 5) ( 6) ( 6) ( 5) ( 6) ( 7) ( 6) ( 7) ( 7) ( 6) ( 7) EOC = log 2 ( E(h)/E( h 2 )) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

49 The Excess Wing model P(t) = 1 + ( 2 iω) α 1 + ( 2 iω) α + 1 iω E(t) 1) Linear multiterm fractional differential equation D t P(t) + 2 α 0Dt α P(t) + 1 ( 1 P(t) = E(t) + α ) 2 0Dt α E(t) ) Convolution quadrature rule h 1 α n=0 h α + 2 α α (1 ξ) α ωn EW ξ n = h 1 α 2 (1 + ξ) α h + 2 α 2 α (1 ξ)α h1 α (1 + ξ) α + (1 ξ) 1 (1 + ξ) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

50 Concluding remarks The Havriliak-Negami dielectric model can be described in the time-domain by Prabhakar derivatives and integrals Prabhakar derivatives and integrals are non standard fractional-order operators depending on several parameters Discretization of Prbhakar operators by means of Lubich s convolution quadratures: direct use of the complex susceptibility General approach: extension to other models: Excess Wing, multichannel Excess Wing, and others... Further developments Improving computational efficiency (memory term) Validation with experimental data (Technical University of Bari) Releasing robust software for solving Maxwell s systems with anomalous dielectric relaxation R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25

51 Some references Garra R., Gorenflo R., Polito F. and Tomovski Z., Hilfer-Prabhakar derivatives and some applications. In: Appl. Math. Comput. 242 (2014) Garra R., Garrappa R., The Prabhakar or three parameter Mittag Leffler function: theory and application. In: CNSNS 56 (2018) Garrappa R., Numerical evaluation of two and three parameter Mittag-Leffler functions. In: SIAM J. Numer. Anal. 53(3) (2015) Garrappa R., Grünwald Letnikov operators for Havriliak Negami fractional relaxation. In: CNSNS 38 (2016) Garrappa R., Mainardi F. and Maione G., Models of dielectric relaxation based on completely monotone functions. In: FCAA 19(5) (2016) Giusti A., Colombaro R., Prabhakar-like fractional viscoelasticity. In: CNSNS 56 (2018) Lubich C. Convolution quadrature and discretized operational calculus (I and II), In: Numer. Math., 52 (1998) Mainardi F. and Garrappa R., On complete monotonicity of the Prabhakar function and non-debye relaxation in dielectrics. In: JCP 293 (2015) R.Garrappa (Univ. of Bari - Italy) Non-standard fractional operators ICERM / 25